Abstact: Let $f$
be a primitive holomorphic cusp form of weight $k$, level $q$, and character $\chi$, and let $L(s,f)$ be its associated $L$-function. I will discuss how to prove quantitative estimates for the number of simple non-trivial zeros of $L(s,f)$ under the assumption of the generalized Riemann Hypothesis. Even assuming GRH, this seems to be the first method capable of proving that infinitely many primitive degree two $L$-functions have an infinitude of simple non-trivial zeros. If there is time, I will discuss an ongoing project where the condition "$L(s,f)$ has infinitude of simple non-trivial zeros" is related to the non-vanishing of a certain average of Dirichlet twists of the derivative of $L(s,f)$ at the central point. Both projects are joint with Nathan Ng (Lethbridge).